研 究 生 : 柯佳男 指導教授 : 陳正宗博士 日 期 : 2007/01/11

1. Null-field integral equation approach for the scattering water wave problems with circular boundaries 研 究 生:柯佳男 指導教授:陳正宗博士 日 期: 2007/01/11

2. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Vector decomposition technique • Linear algebraic equation • Numerical examples • Water wave problem with two circular cylinders • Conclusions

3. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Vector decomposition technique • Linear algebraic equation • Numerical examples • Water wave problem with two circular cylinders • Conclusions

4. Motivation • In many of large ocean structures, the interaction between water waves and arrays of bodies has become increasingly important and work has been done on the subject in recent years. • Exact solutions are limited for the simple case. Numerical solutions are generally required in engineering application. • Subsequently, much of the work due to the scattering of water waves by arrays of bodies have already turned into interesting of evaluating wave forces.

5. Literature review • The null-field (or T-matrix) method • Waterman, 1969 (single scatter) • Peterson and Strom, 1979 (several scatters ) • Wave forces on cylinder arrays • McIver & Evans , 1984 • Shallow water wave • Mingde & Yu, 1987

6. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Vector decomposition technique • Linear algebraic equation • Numerical examples • Water wave problem with two circular cylinders • Conclusions

7. Expansions of fundamental solution (2D) • Laplace problem-- • Helmholtz problem-- x x s O

8. Boundary density discretization • Fourier series expansions - boundary density Fourier series Ex . constant element

9. Adaptive observer system Source point collocation point

10. Linear algebraic equation

11. Degenerate kernel Fourier series Potential Null-field equation Algebraic system Fourier Coefficients Analytical Numerical Flowchart of present method

12. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Vector decomposition technique • Linear algebraic equation • Numerical examples • Water wave problem with two circular cylinders • Conclusions

13. W-Wave Water wave (3-D)

14. (動力自由表面邊界條件) (運動自由表面邊界條件) (側面邊界條件) (底床邊界條件) 邊界條件 I. 合併 可令 II. 滿足

15. Governing equation Governing equation: (Laplace equation) (Helmholtz equation)

16. W-Wave W-Wave Water wave (2-D)

17. Where: Decomposition of coordinate

18. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Vector decomposition technique • Linear algebraic equation • Numerical examples • Water wave problem with two circular cylinders • Conclusions

19. Conclusions • We will calculate the water wave force by using the potential function • We will develop a set of formulation for arbitrary number of circular cylinders with arbitrary radii, proportion and location for the scattering water wave problems

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